$12^2=144$
Here in, $144$ the hundreds digit is 1.
The $1$ has travelled to the units place below in $21$ as well as $441$.
$21^2=441$
What can be said of the $4's$ ?
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asked Jul 17, 2013 at 14:36
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2$\begingroup$ I don't understand. What do you mean, "what can be said of the 4s?" $\endgroup$– Gerry MyersonJul 29, 2013 at 13:02
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1 Answer
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It can be said that they are completely unrelated. Then again, $13^2=169$ and $31^2=961$, so we see the first and last digit switched, the central not moved. Both these results are a consequence of $(10a+b)^2=100a^2 +20ab+b^2$ and work if the digits $a,b$ are such that $0<ab<5$.
You may also want to compare $123000456^2$ and $456000123^2$.
answered Jul 17, 2013 at 14:41
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$\begingroup$ I came across the observation about $12$ and $13$ being such numbers when I was 16 and bored with my homework. For some time, as a child, I thought these were the only two numbers satisfying this property until I learned to program in Common Lisp and wrote a code to collect more of these numbers. Some years later I asked my number theory professor about that, and he helped me a bit, but he was generally unimpressed by this peculiarity. I never got around to ask it here, although I had written several beginning of questions. $\endgroup$– Asaf Karagila ♦Jul 17, 2013 at 14:51
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$\begingroup$ @AsafKaragila I remember the 12 and 13 cases mentioned in an old (early 20th cetury) book with general world facts (history, geography, engineering, ...) in the math section (also when I was a teen) $\endgroup$ Aug 22, 2015 at 20:14
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$\begingroup$ What can I say... took you long enough to reply to my comment! :-) $\endgroup$– Asaf Karagila ♦Aug 22, 2015 at 20:18